14.02.2018 Views

VbvAstE-001

Book Boris V. Vasiliev Astrophysics

Book Boris V. Vasiliev
Astrophysics

SHOW MORE
SHOW LESS

Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

Using the definition of a chemical potential of ideal gas (of particles with spin=1/2)<br />

[12]<br />

[ ( )<br />

Ne 2π<br />

2 3/2 ]<br />

µ e = kT log<br />

(2.8)<br />

2V m ekT<br />

we obtain the full energy of the hot electron gas<br />

[<br />

E e ≈ 3 ( )<br />

2 kT Ne 1 + π3/2 aBe 2 3/2<br />

]<br />

n e , (2.9)<br />

4 kT<br />

where a B =<br />

2<br />

m ee 2<br />

is the Bohr radius.<br />

2.1.3 The correction for correlation of charged particles<br />

in a hot plasma<br />

At high temperatures, the plasma particles are uniformly distributed in space. At this<br />

limit, the energy of ion-electron interaction tends to zero. Some correlation in space<br />

distribution of particles arises as the positively charged particle groups around itself<br />

preferably particles with negative charges and vice versa. It is accepted to estimate<br />

the energy of this correlation by the method developed by Debye-Hükkel for strong<br />

electrolytes [12]. The energy of a charged particle inside plasma is equal to eϕ, where<br />

e is the charge of a particle, and ϕ is the electric potential induced by other particles<br />

on the considered particle.<br />

This potential inside plasma is determined by the Debye law [12]:<br />

where the Debye radius is<br />

r D =<br />

ϕ(r) = e r e− r<br />

r D (2.10)<br />

(<br />

4πe 2<br />

kT<br />

−1/2<br />

∑<br />

n aZa) 2 (2.11)<br />

For small values of ratio<br />

r<br />

r D<br />

, the potential can be expanded into a series<br />

ϕ(r) = e r − e + ... (2.12)<br />

r D<br />

The following terms are converted into zero at r → 0. The first term of this series is<br />

the potential of the considered particle. The second term<br />

√ ( 3/2<br />

E = −e 3 π ∑<br />

N aZa) 2 (2.13)<br />

kT V<br />

a<br />

is a potential induced by other particles of plasma on the charge under consideration.<br />

And so the correlation energy of plasma consisting of N e electrons and (N e/Z) nuclei<br />

with charge Z in volume V is [12]<br />

a<br />

E corr = −e 3 √<br />

πne<br />

kT Z3/2 N e (2.14)<br />

17

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!